Databases are classified as "services". So, if you have a tag rule that applies to services that match a DB rule, then connect that tag to a Management Zone, this should apply to your database services as well.
Correct, but you don't have to define tags for an entity to be included in a management zone - you can just leverage it.
For management zone, you define matching rules for similar to automatic tagging rules. If you already use tags, you can define rules for you management zone to use these tags.